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Math Help - Determine if the following is a vector space

  1. #1
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    Determine if the following is a vector space

    Hello everyone. I'm new here. I have a practice test that I'm working on for my linear algebra class and I sorta get the gist of determining whether or not something is a subspace or not but I'm having trouble with this one:

    I did terrible in Calc 2 so the whole derivative thing isn't too familiar to me.

    Attached Thumbnails Attached Thumbnails Determine if the following is a vector space-0008.jpg  
    Last edited by chickeneaterguy; February 27th 2010 at 11:24 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chickeneaterguy View Post
    Hello everyone. I'm new here. I have a practice test that I'm working on for my linear algebra class and I sorta get the gist of determining whether or not something is a subspace or not but I'm having trouble with this one:

    I did terrible in Calc 2 so the whole derivative thing isn't too familiar to me.




    I also have some T/F questions on it. I have guesses but nothing I know for a fact.

    True/False?:
    1. P2 is a Subspace of P3 (I think it's true)
    2. D[0,1], the set of all differential functions over [0,1] is a subspace of C[0,1] (I think it's true)
    3. {ax^2 + bx + c : b = 0} is a subspace of P2 (I think it's true....or maybe not)
    Obvious things...look for them. (f+g)'(2)=f'(2)+g'(2)=0+0=0. \left(\alpha f\right)(2)=\alpha f'(2)=\alpha \cdot 0=0. etc.
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    Obvious things...look for them. (f+g)'(2)=f'(2)+g'(2)=0+0=0. \left(\alpha f\right)(2)=\alpha f'(2)=\alpha \cdot 0=0. etc.
    Thanks for the prompt reply.

    So I don't even need to bother with derivatives or anything?

    EDIT: nvm...I realized how obvious it is. Gosh...that's ridiculously easy.
    Last edited by chickeneaterguy; February 27th 2010 at 01:05 AM.
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    Quote Originally Posted by chickeneaterguy View Post
    Thanks for the prompt reply.

    So I don't even need to bother with derivatives or anything?

    EDIT: nvm...I realized how obvious it is. Gosh...that's ridiculously easy.
    How would you prove :

    for all ,f,gεV ..........f+g=g+f ??
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  5. #5
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    As you would prove any such fundamental statement- from the definition:

    f+ g is defined as the function such that (f+g)(x)= f(x)+ g(x).
    Similarly, g+ f is defined as the function such that (g+f)(x)= g(x)+ f(x).

    Saying that f+ g= g+ f is just saying that, for all x, f(x)+ g(x)= g(x)+ f(x) - and that's true because f(x) and g(x) are numbers and addition of numbers is commutative.
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